Description
True Galois-qudit stabilizer code constructed from evaluation AG codes via the Galois-qudit Hermitian construction or the Galois-qudit CSS construction.Rate
Quantum AG codes can be asymptotically good [1,2]. There exist three such families [3–5] that admit a diagonal transversal gate at the third level of the Clifford hierarchy.Magic
By defining a generalization of triorthogonal matrices to Galois qudits of dimension \(q=2^m\), one can construct an asymptotically good family of quantum AG codes that admits a diagonal transversal gate at the third level of the Clifford hierarchy and attains a zero magic-state yield parameter, \(\gamma = 0\) [3]. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of \(m\) qubits; see [4–6,8,9][7; Sec. 5.3]. Two other asymptotically good families [4,5] admit a transversal \(CCZ\) gate (a different diagonal gate at the third level of the Clifford hierarchy) and achieve \(\gamma \to 0\) with constant alphabet size.Encoding
Encoding defined in Ref. [2] uses a technique from Ref. [10] to encode quantum stabilizer codes.Transversal Gates
There exist three asymptotically good code families [3–5] that admit a diagonal transversal gate at the third level of the Clifford hierarchy.By decomposing each Galois qudit into a Kronecker product of qubits, the family of Ref. [5] yields an explicit asymptotically good qubit CSS code family with parameters \([[N,K=\Theta(N),D=\Theta(N)]]\) on which \(\overline{CCZ}^{\otimes K}\) is realized by a transversal application of physical \(CCZ\) gates on a constant fraction of qubits.There exists an asymptotically good code family that admits three-Galois-qudit non-Clifford gates for any three logical Galois qudits [11].Cousins
- Evaluation AG code— Quantum AG codes are quantum analogues of evaluation AG codes.
- Triorthogonal code— By defining a generalization of triorthogonal matrices to Galois qudits of dimension \(q=2^m\), one can construct an asymptotically good family of quantum AG codes that admits a diagonal transversal gate at the third level of the Clifford hierarchy and attains a zero magic-state yield parameter, \(\gamma = 0\) [3]. This code can be treated as a qubit code by decomposing each Galois qudit into a Kronecker product of \(m\) qubits; see [4–6,8,9][7; Sec. 5.3]. Two other asymptotically good families [4,5] admit a transversal \(CCZ\) gate (a different diagonal gate at the third level of the Clifford hierarchy) and achieve \(\gamma \to 0\) with constant alphabet size.
- Tsfasman-Vladut-Zink (TVZ) code— The AG codes used in an asymptotically good construction of quantum AG codes with non-Clifford transversal gates [4] are those of the TVZ codes.
- Elliptic code— Elliptic codes can be used to construct quantum AG codes [12].
Primary Hierarchy
Parents
Quantum AG codes can be constructed via the Galois-qudit CSS construction or the Galois-qudit Hermitian construction.
Quantum AG code
Children
Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction from GRS codes, which are evaluation AG codes.
References
- [1]
- H. Chen, “Some Good Error-Correcting Codes from Algebraic-Geometric Codes”, (2001) arXiv:quant-ph/0107102
- [2]
- R. Matsumoto, “Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes”, IEEE Transactions on Information Theory 48, 2122 (2002) arXiv:quant-ph/0107129 DOI
- [3]
- A. Wills, M.-H. Hsieh, and H. Yamasaki, “Constant-Overhead Magic State Distillation”, (2024) arXiv:2408.07764
- [4]
- L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
- [5]
- Q. T. Nguyen, “Good binary quantum codes with transversal CCZ gate”, (2024) arXiv:2408.10140
- [6]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [7]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
- [8]
- D. Gottesman, Surviving as a Quantum Computer in a Classical World (2024) URL
- [9]
- M. Heinrich. On stabiliser techniques and their application to simulation and certification of quantum devices. PhD thesis, Universität zu Köln, 2021
- [10]
- A. Ashikhmin and E. Knill, “Nonbinary Quantum Stabilizer Codes”, (2000) arXiv:quant-ph/0005008
- [11]
- Z. He, V. Vaikuntanathan, A. Wills, and R. Y. Zhang, “Quantum Codes with Addressable and Transversal Non-Clifford Gates”, (2025) arXiv:2502.01864
- [12]
- L. Sok, “New families of quantum stabilizer codes from Hermitian self-orthogonal algebraic geometry codes”, (2021) arXiv:2110.00769
Page edit log
- Victor V. Albert (2024-08-23) — most recent
Cite as:
“Quantum AG code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quantum_ag